Normalized defining polynomial
\( x^{10} - 4x^{9} - 9x^{8} + 46x^{7} + 5x^{6} - 139x^{5} + 59x^{4} + 106x^{3} - 40x^{2} - 17x - 1 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[10, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(1056004125376537\) \(\medspace = 101833^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(31.80\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
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Galois root discriminant: | $101833^{1/2}\approx 319.11283270968596$ | ||
Ramified primes: | \(101833\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{101833}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2237}a^{9}+\frac{707}{2237}a^{8}-\frac{657}{2237}a^{7}+\frac{452}{2237}a^{6}-\frac{751}{2237}a^{5}+\frac{543}{2237}a^{4}-\frac{869}{2237}a^{3}-\frac{341}{2237}a^{2}-\frac{895}{2237}a-\frac{1054}{2237}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{432}{2237}a^{9}-\frac{3282}{2237}a^{8}+\frac{275}{2237}a^{7}+\frac{38674}{2237}a^{6}-\frac{42570}{2237}a^{5}-\frac{118870}{2237}a^{4}+\frac{143576}{2237}a^{3}+\frac{85336}{2237}a^{2}-\frac{77934}{2237}a-\frac{5691}{2237}$, $\frac{1016}{2237}a^{9}-\frac{2002}{2237}a^{8}-\frac{14308}{2237}a^{7}+\frac{20780}{2237}a^{6}+\frac{60200}{2237}a^{5}-\frac{52302}{2237}a^{4}-\frac{84295}{2237}a^{3}+\frac{24886}{2237}a^{2}+\frac{32457}{2237}a+\frac{2896}{2237}$, $\frac{1016}{2237}a^{9}-\frac{2002}{2237}a^{8}-\frac{14308}{2237}a^{7}+\frac{20780}{2237}a^{6}+\frac{60200}{2237}a^{5}-\frac{52302}{2237}a^{4}-\frac{84295}{2237}a^{3}+\frac{24886}{2237}a^{2}+\frac{32457}{2237}a+\frac{659}{2237}$, $\frac{432}{2237}a^{9}-\frac{3282}{2237}a^{8}+\frac{275}{2237}a^{7}+\frac{38674}{2237}a^{6}-\frac{42570}{2237}a^{5}-\frac{118870}{2237}a^{4}+\frac{143576}{2237}a^{3}+\frac{85336}{2237}a^{2}-\frac{77934}{2237}a-\frac{7928}{2237}$, $\frac{1721}{2237}a^{9}-\frac{4655}{2237}a^{8}-\frac{21145}{2237}a^{7}+\frac{50867}{2237}a^{6}+\frac{69862}{2237}a^{5}-\frac{139257}{2237}a^{4}-\frac{63869}{2237}a^{3}+\frac{77528}{2237}a^{2}+\frac{21131}{2237}a+\frac{2510}{2237}$, $a$, $\frac{1986}{2237}a^{9}-\frac{7445}{2237}a^{8}-\frac{18527}{2237}a^{7}+\frac{83404}{2237}a^{6}+\frac{16252}{2237}a^{5}-\frac{236958}{2237}a^{4}+\frac{104032}{2237}a^{3}+\frac{145990}{2237}a^{2}-\frac{81824}{2237}a-\frac{6123}{2237}$, $\frac{1524}{2237}a^{9}-\frac{3003}{2237}a^{8}-\frac{21462}{2237}a^{7}+\frac{31170}{2237}a^{6}+\frac{90300}{2237}a^{5}-\frac{78453}{2237}a^{4}-\frac{125324}{2237}a^{3}+\frac{37329}{2237}a^{2}+\frac{40856}{2237}a+\frac{2107}{2237}$, $\frac{668}{2237}a^{9}-\frac{1968}{2237}a^{8}-\frac{7135}{2237}a^{7}+\frac{20074}{2237}a^{6}+\frac{15079}{2237}a^{5}-\frac{44410}{2237}a^{4}+\frac{7839}{2237}a^{3}+\frac{2623}{2237}a^{2}-\frac{5055}{2237}a+\frac{583}{2237}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 14721.0314836 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{0}\cdot 14721.0314836 \cdot 1}{2\cdot\sqrt{1056004125376537}}\cr\approx \mathstrut & 0.231939862353 \end{aligned}\]
Galois group
A non-solvable group of order 120 |
The 7 conjugacy class representatives for $S_5$ |
Character table for $S_5$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 5 sibling: | 5.5.101833.1 |
Degree 6 sibling: | 6.6.1056004125376537.1 |
Degree 10 sibling: | data not computed |
Degree 12 sibling: | data not computed |
Degree 15 sibling: | data not computed |
Degree 20 siblings: | data not computed |
Degree 24 sibling: | data not computed |
Degree 30 siblings: | data not computed |
Degree 40 sibling: | data not computed |
Minimal sibling: | 5.5.101833.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.3.0.1}{3} }^{3}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.5.0.1}{5} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(101833\) | $\Q_{101833}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{101833}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{101833}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{101833}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |